I misread your post. All spending starts in the House. In order to get bills passed, however, it has to have approval from the Senate and signed by the President. So, yes, they don’t have to put forth appropriations, but, constitutionally, that is their job.
It’s their job, and their responsibility. They’re the ones who deserve the blame or credit for what funds are, or are not, budgeted. That means what they decide deserves first priority.
And the job of the President and Senate is to veto any spending they feel is unwise. It is NOT their job or responsibility to demand spending that the House is not willing to approve.